On Blowup solutions to the focusing masscritical nonlinear fractional Schrödinger equation
Abstract.
In this paper we study dynamical properties of blowup solutions to the focusing masscritical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the concentration and the limiting profile with minimal mass of blowup solutions.
Key words and phrases:
Nonlinear fractional Schrödinger equation; Blowup; Concentration; Limiting profile2010 Mathematics Subject Classification:
35B44, 35Q551. Introduction
Consider the Cauchy problem for nonlinear fractional Schrödinger equations
(1.1) 
where is a complex valued function defined on , and . The parameter (or ) corresponds to the defocusing (or focusing) case. The operator is the fractional Laplacian which is the Fourier multiplier by . The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was discovered by Laskin [17] as a result of extending the Feynmann path integral, from the Brownianlike to Lévylike quantum mechanical paths. The equation enjoys the scaling invariance
A computation shows
We thus define the critical exponent
(1.2) 
The equation also enjoys the formal conservation laws for the mass and the energy:
The local wellposedness for in Sobolev spaces was studied in [13] (see also [5] for fractional Hartree equations). Note that the unitary group enjoys several types of Strichartz estimates (see e.g. [4] or [7] for Strichartz estimates with nonradial data; and [12], [16] or [2] for Strichartz estimates with radially symmetric data; and [8] or [3] for weighted Strichartz estimates). For nonradial data, these Strichartz estimates have a loss of derivatives. This makes the study of local wellposedness more difficult and leads to a weak local theory comparing to the standard nonlinear Schrödinger equation (see e.g. [13] or [7]). One can remove the loss of derivatives in Strichartz estimates by considering radially symmetric initial data. However, these Strichartz estimates without loss of derivatives require an restriction on the validity of , that is . We refer the reader to Section 2 for more details about Strichartz estimates and the local wellposedness in for .
Recently, BoulengerHimmelsbachLenzmann [1] proved blowup criteria for radial solutions to the focusing . More precisely, they proved the following:
Theorem 1.1 (Blowup criteria [1]).
Let , and . Let be radial such that the corresponding solution to the focusing defines on the maximal time interval .

Masscritical case, i.e. or : If , then the solution either blows up in finite time, i.e. or blows up in infinite time, i.e. and
with some and that depend only on and .

Mass and energy intercritical case, i.e. or : If and
where is the unique (modulo symmetries) positive radial solution to the elliptic equation
then the solution blows up in finite time, i.e. .

Energycritical case, i.e. or : If and
where is the unique (modulo symmetries) positive radial solution to the elliptic equation
the the solution blows up in finite time, i.e. .
In this paper we are interested in dynamical properties of blowup solutions in for the focusing masscritical nonlinear fractional Schrödinger equation, i.e. , and in . Before entering some details of our results, let us recall known results about blowup solutions in for the focusing masscritical nonlinear Schrödinger equation
The existence of blowup solutions in for (mNLS) was firstly proved by Glassey [11], where the author showed that for any negative initial data satisfying , the corresponding solution blows up in finite time. OgawaTsutsumi [26, 27] showed the existence of blowup solutions for negative radial data in dimensions and for any negative data (without radially symmetry) in the one dimensional case. The study of blowup solution to (mNLS) is connected to the notion of ground state which is the unique (up to symmetries) positive radial solution to the elliptic equation
By the variational characteristic of the ground state, Weinstein [30] showed the structure and formation of singularity of the minimal mass blowup solution, i.e. . He proved that the blowup solution remains close to the ground state up to scaling and phase parameters, and also translation in the nonradial case. MerleTsutsumi [18], Tsutsumi [29] and Nava [25] proved the concentration of blowup solutions by using the variational characterization of ground state, that is, there exists such that for all ,
where is the blowup time. Merle [19, 20] used the conformal invariance and compactness argument to characterize the finite time blowup solutions with minimal mass. More precisely, he proved that up to symmetries of the equation, the only finite time blowup solution with minimal mass is the pseudoconformal transformation of the ground state. HmidiKeraani [14] gave a simplified proof of the characterization of blowup solutions with minimal mass of Merle by means of the profile decomposition and a refined compactness lemma. MerleRaphaël [21, 22, 23] established sharp blowup rates, profiles of blowup solutions by the help of spectral properties.
As for (mNLS), the study of blowup solution to the focusing masscritical nonlinear fractional Schrödinger equation is closely related to the notion of ground state which is the unique (modulo symmetries) positive radial solution of the elliptic equation
(1.3) 
The existence and uniqueness (up to symmetries) of ground state for were recently shown in [9] and [10]. In [1, 10], the authors showed the sharp GagliardoNirenberg inequality
(1.4) 
where
Using this sharp GagliardoNirenberg inequality together with the conservation of mass and energy, it is easy to see that if satisfies
then the corresponding solution exists globally in time. This implies that is the critical mass for the formation of singularities.
To study blowup dynamics for data in , we establish the profile decomposition for bounded sequences in in the same spirit of [14]. With the help of this profile decomposition, we prove a compactness lemma related to the focusing masscritical (NLFS).
Theorem 1.2 (Compactness lemma).
Let and . Let be a bounded sequence in such that
Then there exists a sequence in such that up to a subsequence,
for some satisfying
(1.5) 
where is the unique solution to the elliptic equation .
Note that the lower bound on the norm of is optimal. Indeed, if we take , then we get the identity.
As a consequence of this compactness lemma, we show that the norm of blowup solutions must concentrate by an amount which is bounded from below by at the blowup time. Finally, we show the limiting profile of blowup solutions with minimal mass . More precisely, we show that up to symmetries of the equation, the ground state is the profile for blowup solutions with minimal mass.
The paper is oganized as follows. In Section 2, we recall Strichartz estimates for the fractional Schrödinger equation and the local wellposedness for in nonradial and radial initial data. In Section 3, we show the profile decomposition for bounded sequences in and prove a compactness lemma related to the focusing masscritical . The concentration of blowup solutions is proved in Section 4. Finally, we show the limiting profile of blowup solutions with minimal mass in Section 5.
2. Preliminaries
2.1. Strichartz estimates
In this subsection, we recall Strichartz estimates for the fractional Schrödinger equation. Let and . We define the Strichartz norm
with a usual modification when either or are infinity. We have threetypes of Strichartz estimates for the fractional Schrödinger equation:

For radially symmetric data (see e.g. [16], [12] or [2]): the estimates and hold true for and satisfy the radial Schödinger admissible condition:
Note that the admissible condition
(2.3) (2.4) where and are radially symmetric and satisfy the fractional admissible condition,
(2.5) These Strichartz estimates with no loss of derivatives allow us to give a similar local wellposedness result as for the nonlinear Schrödinger equation (see again Subsection 2.3).

Weighted Strichartz estimates (see e.g. [8] or [3]): for and ,
(2.6) and for , and
(2.7) Here with is the LaplaceBeltrami operator on the unit sphere . Here we use the notation
These weighted estimates are important to show the wellposedness below at least for the fractional Hartree equation (see [5]).
2.2. Nonlinear estimates
We recall the following fractional chain rule which is needed in the local wellposedness for .
2.3. Local wellposedness in
In this section, we recall the local wellposedness in the energy space for . As mentioned in the introduction, we will separate two cases: nonradial initial data and radially symmetric initial data.
Nonradial initial data.
Proposition 2.2 (Nonradial local theory [13, 7]).
Let and be such that
(2.8) 
Then for all , there exist and a unique solution to satisfying
for some when and some when . Moreover, the following properties hold:

If , then as .

There is conservation of mass, i.e. for all .

There is conservation of energy, i.e. for all .
The proof of this result is based on Strichartz estimates and the contraction mapping argument. The loss of derivatives in Strichartz estimates can be compensated for by using the Sobolev embedding. We refer the reader to [13] or [7] for more details.
Remark 2.3.
It follows from and that the local wellposedness for non radial data in is available only for
(2.9) 
In particular, in the masscritical case , the is locally wellposed in with
Proposition 2.4 (Nonradial global existence [7]).
Let and be as in . Then for any , the solution to given in Proposition 2.2 can be extended to the whole if one of the following conditions is satisfied:

,

and ,

, and is small,

and is small.
Proof.
The case follows easily from the blowup alternative together with the conservation of mass and energy. The case and follows from the GagliardoNirenberg inequality (see e.g. [28, Appendix]). Indeed, by GagliardoNirenberg inequality and the mass conservation,
The conservation of energy then implies
If , then and hence . This combined with the conservation of mass yield the boundedness of for any belongs to the existence time. The blowup alternative gives the global existence. The case and small is treated similarly. It remains to treat the case and is small. Thanks to the Sobolev embedding with
This shows in particular that is small if is small. Therefore,
This shows that is bounded and the proof is complete. ∎
Radial initial data.
Thanks to Strichartz estimates without loss of derivatives in the radial case, we have the following result.
Proposition 2.5 (Radial local theory).
Let and and . Let
(2.10) 
Then for any radial, there exist and a unique solution to satisfying
Moreover, the following properties hold:

If , then as .

for any fractional admissible pair .

There is conservation of mass, i.e. for all .

There is conservation of energy, i.e. for all .
Proof.
It is easy to check that satisfies the fractional admissible condition . We choose so that
(2.11) 
We see that
(2.12) 
The later fact gives the Sobolev embedding . Let us now consider
equipped with the distance
where and to be chosen later. By Duhamel’s formula, it suffices to prove that the functional
is a contraction on . By radial Strichartz estimates and ,
The fractional chain rule given in Lemma 2.1 and the Hölder inequality give
Similarly,
This shows that for all , there exists independent of and such that
If we set and choose so that
then is a strict contraction on . This proves the existence of solution . By radial Strichartz estimates, we see that for any fractional admissible pairs . The blowup alternative follows easily since the existence time depends only on the norm of initial data. The proof is complete. ∎
As in Proposition 2.4, we have the following criteria for global existence of radial solutions in .
Proposition 2.6 (Radial global existence).
Let and . Then for any radial, the solution to given in Proposition 2.5 can be extended to the whole if one of the following conditions is satisfied:

,

and ,

, and is small,

and is small.
Combining the local wellposedness for nonradial and radial initial data, we obtain the following summary.
LWP  

nonradial  
nonradial  
nonradial  
radial  
nonradial  
radial 
Corollary 2.7 (Blowup rate).
Proof.
We follow the argument of MerleRaphael [24]. Let be fixed. We define
with to be chosen shortly. We see that is welldefined for
Moreover, solves
A direct computation shows
Since , we choose so that . Thanks to the local theory, there exists such that is defined on . This shows that
The proof is complete. ∎
3. Profile decomposition
In this subsection, we use the profile decomposition for bounded consequences in to show a compactness lemma related to the focusing masscritical .
Theorem 3.1 (Profile decomposition).
Let and . Let be a bounded sequence in . Then there exist a subsequence of (still denoted ), a family of sequences in and a sequence of functions such that

for every ,
(3.1) 
for every and every ,
with
(3.2) for every , where
Moreover,
(3.3) (3.4) as .
Proof.
The proof is similar to the one given by HmidiKeraani [14, Proposition 3.1]. For reader’s convenience, we recall some details. Since is a Hilbert space, we denote the set of functions obtained as weak limits of sequences of the translated with a sequence in . Denote